CISC 871: Fuzzy Logic and FuzzyAlgorithms

Term Paper

Exploring Space with Fuzzy Logic

Author:Brenna Douglas (6302386)

Professor:Dr. Robin Dawes

April 28, 2012

Abstract

The applications of fuzzy logic and fuzzy algorithms to the classification of

celestial objects will be explored through various prominent papers in the field.

Specifically, these papers will address issues concerning star and galaxy separation,

classification of stars into their spectral types, classification of galaxies into the

main galaxy types, and comparing the results to their neural network counterparts.

Additionally, the task of improving astronomical image processing through the

use of fuzzy logic based algorithms will be explored. A brief discussion will follow

regarding the application and practicality of fuzzy logic to astronomical data mining

and astronomical image processing.

1 Introduction

In the past few years alone, the amount of data that has been collected for as-

tronomical images has been growing exponentially. Ideally, these images will be

used to answer fundamental questions about the universe, such as its composition,

the existence of terrestrial planets with the possibility of life, or finding asteroids

that pose a threat to the Earth. In addition to an increase in number of robotic

telescopes, many new sky surveys, both ground-based and space-based, have been

launched or are in the planning phases.

The Sloan Digital Sky Survey, which began collecting data in 2000, has since

collected spectral data for more than a million objects and mapped more than 35%

of the sky [3]. Pan-STARRS is another ambitious venture that went online in 2008

with the goal of covering the entire sky every 10 days while collecting 13 terabytes

of data per night [7]. Additionally, space based missions, such as Kepler and JWST

([1], [2]), are also continuously adding to the astronomical amount of data already

gathered.

With this vast amount of data pouring in daily, the task of identifying specific

celestial objects is more than just an overwhelming task for humans; it is imprac-

tical. Most of our knowledge of galaxy classification is still based on the work of

several dedicated observers. For example, one study consisted of three people inde-

1

Figure 1: The Hubble sequence for galaxy classification. Invented by Edwin Hubble in1926, the scheme divides galaxies into three classes based on appearances.

pendently classifying 2253 objects, with the final classification taken as the mean

of the three observations [4]. Galaxy Zoo is yet another attempt to classify celes-

tial objects, but makes use of the power of the internet. Upon its launch in 2007,

Galaxy Zoo consisted of a data set of one million from the Sloan Digital Sky Survey.

Galaxy Zoo users were then given the simple task of classifying images of galaxies

into two types, and were given the necessary information to do this. The results

were great; each object had multiple classification sources and it was shown that

the results were as good as those completed by professional astronomers [10].

These both provide excellent databases of classified objects, however, the size of

currently classified object pales in comparison to the amount of data continuously

being recorded. Therefore, it is apparent that an automated classification system

that performs as well as a professional astronomer is needed. As this is essentially

a data mining problem, many people have already applied various data mining

techniques to this task. Here we will explore the application of fuzzy logic and

algorithms to this problem.

2 Separating Stars and Galaxies

The night sky contains an abundance of different kinds of objects. There are stars,

planets, galaxies, clusters, black holes, and nebulae, to name a few. Even within

these categories, each object has its own classification system. For example, galaxies

can be classified into spiral, barred-spiral, irregular, or elliptical galaxies. Figure 1

is a common, transparent version of a classification scheme for galaxies, but not the

2

Figure 2: The Hertzsprung-Russell diagram for star classification demonstrates the rela-tionship between spectral types, temperature and absolute magnitudes of stars.

only one. The Hertzsprung-Russell Diagram, as seen in figure 2, demonstrates one

of the more common diagrams for star classification. There are other classifications

schemes, for both stars and galaxies, as well as the other types of celestial objects,

and for most cases, classification means taking these objects and categorizing them

into discrete sets. Considering the vast amount of data as previous discussed,

classifying a random object into its correct class becomes a daunting task.

Therefore, many people have taken a more simplistic approach for classification.

In 2000, Mahonen and Frantti attempted to tackle this challenge by simplifying it

to a problem of distinguishing stars from galaxies only. Previous works had already

made attempts to use various types of neural networks for star/galaxy separation,

but Mahonen and Frantti take a different approach and demonstrate an automatic

classification system which uses fuzzy set reasoning [12].

[12] used an extension of the k-means clustering algorithm, the fuzzy c-means

clustering algorithm. Similar to the k-means clustering algorithm, the fuzzy c-

means algorithm attempts to create c cluster centers in the space, and then mea-

sures the distance of each point to the center of each cluster. Unlike the k-means

clustering algorithm, the fuzzy c-means algorithm computes the degree to which

each point belongs to a cluster. This allows points farther away from the center to

be in the cluster to a lesser degree than those points close to the center.

Measurements for each image, such as elliplicity, average transmission over im-

age, and the image gradients, were computed. In total, fourteen measurements were

3

Figure 3: Taken from [12], the results of the fuzzy classifier (Fuzzy), a back-propagationnetwork (BP), and self-organizing maps (SOM). Values are in percentages. Size representsthe diamater of the object.

made and used as the parameters to define each image. Using a data set consisting

of 5528 stars and 3717 galaxies, [12] compared the results of their Fuzzy classifier to

two neural network approaches (using back propagation and self-organizing maps)

over various subsets of the data. The results can be seen in figure 3.

Although the results are not quite as good as those from neural networks, it

is pointed out that the fuzzy classifier has its own advantages. The output of

the classifier can be used directly as an estimate of how reliable the classification

is. For examples, classification memberships lie within the range [0,1], and those

classifications which are in the middle of the range (around 0.5) are considered very

unreliable. These objects can then be sent for more sensitive processing, whether

it be a different classification system, or a human observer.

Additionally, the fuzzy classifier can be used for preclassification for a different

classification system. The paper suggests using it as a preclassifier for their neural

networks, but discusses one of the key issues is to determine how to weight the

output from the fuzzy preclassification.

3 Using a Different Separation Approach

The results from [12] demonstrate the use of fuzzy logic for star/galaxy separation,

but is not the only paper to do so. Longo et al. also aimed to create a classifica-

tion system based on unsupervised learning to perform star/galaxy separation [11].

However, their approach was different than that of the previous paper. Their idea

is to first determine two prototypes for the stars and galaxies classes to be used

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as references for each catalog object. Through fuzzy logic, the degree of similarity

with respect two each prototype can then be computed.

To first compute the prototypes, Self-Organizing Maps ([8]) were used. A Self-

Organizing Map is essentially a type of neural network, and in this usage, the first

level is composed with as many nodes as there are object features. The second level

is composed of two nodes, one for stars, and one for galaxies. Therefore, the final

weights represent the prototypes for each class. Object features were measured

from various photometic and astrometric properties of each image, and the seven

most significant features were used.

Once the prototypes are created, the similarities of objects in the catalog to

the prototypes is computed. To compute the similarity, the paper uses Lukasiewicz

algebra. In Lukasiewicz algebra, the binary operations→ and ⊗ (the residuum and

t-norm, respectively), are defined by

x→ y = min{1,√

1− x+ y} (1)

x⊗ y =√max{0, x+ y − 1} (2)

From this, we can use the bi-residuum to interpret fuzzy logic equivalence. The

bi-residuum can be defined as

x↔ y = (x→ y) ∧ (x← y) (3)

Using Lukasiewicz algebra, this becomes

x↔ y = 1−max(x, y) +min(x, y) (4)

Using µX(x) to represent the membership of element x in set X, then fuzzy simi-

larity S can be described by

S(x, y) = µX(x)↔ µX(y) (5)

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Then, to compute the total fuzzy similarity over all n features, we use the equation

S(x, y) =1

n

n∑i=1

Si(x, y) (6)

Using 6, we now have a means to compute the similarity between the prototypes.

Once computed, we have a measure of similarity between the objects in the catalog

and the two prototypes. The final step for classification is to defuzzify. That is,

an object from the catalog is classified as a member of the star class or the galaxy

class based on the maximum similarity with respect to each prototype.

For the actual experiment, 10,000 objects were randomly chosen from a catalog

of 231,000 labeled star and galaxy objects. They obtained a 77.38% classification,

with 2867 galaxies and 4871 stars correctly classified, in incorrectly classified 590

star objects and 1672 galaxy objects. These results at least perform better than

chance, however, for practical use it is not reasonable. Additionally, the method

from the previous paper seemed to perform much better. Nevertheless, these were

preliminary results and the potential for improvement exists.

4 Galaxy Morphology

While the previous papers aimed to simply separate star and galaxy objects into

their respective classes, others attempted to broaden the abilities of their algo-

rithms. In a more recent paper, Gauci et al. aspired to design an intelligent

algorithm with the same accuracy as humans that could distinguish between spiral

galaxies, elliptical galaxies, and stars or unknown objects. Here, the input object

requirements are relaxed, as it is not necessary for the object to be a star or galaxy

object. While classifying galaxies into elliptical or spiral galaxies is a bit simplified

(see Figure 1 as an example of more detailed classification), it clearly is much more

complex than simply determining a galaxy.

As mentioned before, most of our current knowledge of galaxy classification is

based on the work of several dedicated observers who visual inspect and catalog

thousands of galaxies. The Galaxy Zoo project has provided quite an extensive

collection of about a million labeled objects. This is a great accomplishment and

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Figure 4: Taken from [5], the results of the fuzzy inference system.

extremely useful to the field of astronomy and astrophysics, as the results can be

used for data sets for experiments, such as new machine learning algorithms for

astronomical data.

A million may seem like a large number, but the distribution of objects is not

even. We have many bright, old galaxies, but we have a limited number of dim

galaxies, dark-matter galaxies, or dwarf galaxies, simply because they are harder to

see. Likewise, young galaxies and proto-galaxies that formed soon after the big bang

are not well documented compared to common spiral or elliptical galaxies, either.

Additionally, even for humans it can be difficult to distinguish what classification to

use for a galaxy. For example, when two galaxies collide, depending on the extent of

the current collision, it might be difficult to distinguish if they should be classified

as two separate galaxies, or one galaxy.

While we may not have as many examples of these, they are certainly interest-

ing and obviously offer a lot of useful information about the universe. Therefore,

although a million objects is a lot, there are a lot of categories of galaxy classes

that are completely underrepresented. It then becomes apparent that intelligent

algorithms for automated classification is a huge challenge and being able to classify

galaxies into two subcategories is a great step in the right direction.

[5] compared results from a fuzzy inference system and results from various

decision tree algorithms. Specifically, they used Random Forests, the Classification

and Regression Tree (CART) scheme, and the C4.5 decision tree learner. They

took their training and testing samples from Galaxy Zoo catalog, and computed

various photometric and spectral attributes for each object, using a total of thirteen

attributes. Such measurements include (but not limited to) the DeVancouleurs fit

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Figure 5: Taken from [5], the results of the various decision tree algorithms used.

axis ratio, concentration, star log likelihood, and adaptive fourth moment.

Of interest to this paper is their use of fuzzy logic. They made use of a fuzzy

inference system, where if-then rules that deal with fuzzy consequents and fuzzy

antecedents are defined. First, the degree of truth for each antecedent is computed,

which then can be used to compute the truth of each consequent. The resulting

consequents are then weighted and combined by standard logical operators. The

standard min and max functions are used as the t-norms and s-norms, and the final

result is defuzzified to obtain one value. Disappointingly, the paper did not include

the inference rules used. Although it would probably require a lot of background

knowledge of properties of each class (elliptical galaxies, spiral galaxies, stars),

examples would have been appreciated.

The results for the fuzzy inference system can be seen in figure 4. It was able

to correctly classify 96% of the elliptical galaxies and 92% of spiral galaxies, which

is good news. However, it was only able to classify 55% of the unknowns correctly.

Looking at the confusion matrix, it is apparent that for both galaxy types, they

were rarely classified as unknowns incorrectly. The unknowns did slightly better

than chance (chance would be 33%), and the overall classification had an accuracy

of 94%. Depending on the goals for the output, this could be a reasonable result.

For comparison, the corresponding decision tree algorithm results can be seen

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in figure 5. We can see that the decision trees have an overall accuracy of at least

96%. Additionally, unknowns are all classified correctly at least 83% of the time.

From these results we could suspect that decision trees provide a better algorithm

than fuzzy inference trees.

However, it is important to note that these calculations were based off of pa-

rameters that were calculated in the i-band. That is, they were measured in the

infrared range. The paper also performed calculations in the r-band (the red range)

input parameters, as well as spectra input parameters, and classification using the

various decision tree methods were performed with these input parameters. It is

interesting that they did not use these measurements towards the fuzzy inference

system. Perhaps it was due to only a slight improvement in overall classification

(about 0.5%) for decision trees, although no explanation was given.

It seems as though there is a lot of room for improvement and growth in this

particular application. Indeed, the paper does discuss that further study could be

apply to the fuzzy inference system, to both improve the rules used and the final re-

sults. It was not explained why the spectral parameters and the r-band parameters

were not used, but this is another area that could be explored. Additionally, like

many other papers, they suggest that this technique could be used as a preliminary

classification technique. Therefore, it would suggest that hope is not yet lost for

the application of fuzzy logic and astronomical object classification.

5 Fuzzy Reasoning and Stars

The previous papers are essentially various ways to separate stars and galaxies,

but do not get much more complex. Rodriguez et al. look at the problem of

classification differently. Their goal is to create an intelligent system for the analysis

and classification of low-resolution optical spectra of super giant, giant and dwarf

stars, with luminosity levels I, III, and V, respectively [14].

Refer to 2 to see these classifications. Essentially their goal is to determine main

sequence stars from giants and from super giants, but ignore the white dwarfs. A

luminosity level of II is for stars classified as ”bright giants” and a luminosity level

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of IV is for stars classified as ”sub-giants” (these are not in figure 2). Both of

these classes straddle the boundary between super giants and giants, and giants

and main sequence stars. It makes sense to avoid attempting to include them in

the classification in the beginning stages of start classification, as their classification

can be tricky, even for an expert.

Using stellar spectroscopy, the physical conditions (temperature, pressure, etc.)

and chemical components of stars can be measured. A stellar spectrum for a star can

be collected by using a telescope with the appropriate spectrographs and detectors

mounted. In addition to the luminosity level classes, stars can be put into classes

based on their temperature using the Morgan-Keenan system [9]. These classes

are O, B, A, F, G, K, and M, with O class stars being the hottest, and M class

stars being the coolest. (This can also be seen in figure 2). Stars that belong to

specific temperature and luminosity classes will produce different spectrographs.

Using luminosity, temperature, and spectral type, a star can be classified into the

classes super giant, giant and dwarf (or main sequence) stars.

Again, as is the case with most astronomical data mining, classifications based

on the spectrographs were mostly carried out by hand by dedicated experts, and

it is very time-consuming and requires a lot of human resources. Therefore, the

authors created a process to simulate the behaviour of human experts using fuzzy

and knowledge-based reasoning based on a set of uncertain classification criteria

derived from previous experience. From the spectral features, ten molecular bands

and nine emission/absorption lines and their relationships were found to be the

main reasoning criteria used by human experts for manual processing of stellar

classification.

Typically, human experts visually observe the spectral features and obtain a

preliminary classification, which includes the spectral type, the luminosity and

the global group (early, intermediate, and late). From there, they compare each

spectrum to the reference catalog to obtain the spectral sub-type. Sometimes, it is

difficult to obtain the spectral sub-type interval, in which case they are classified

with two numbers to indicate the sub-type interval. Because the human reasoning

itself includes uncertainty and imprecision, it is easy to see that fuzzy logic could

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Figure 6: Taken from [14], an example fuzzy variable with its corresponding fuzzy set for’Global clssification in luminosity I’.

Figure 7: Taken from [14]. The performance of the automatic classification techniquesand two human experts.

be of use.

Rodriguez et al. used fuzzy if-then production rules to model the reasoning

followed by experts in the field. The antecedent conditions of these rules refer to

the values of the measured parameters (which are stored in a current facts base).

The consequents allude to the three levels of spectral classification. Fuzzy sets and

membership functions were determined by the values of the spectral features in the

guiding catalog spectra. That is, they defined as many fuzzy variables as there were

classification levels for each luminosity class. An example can be seen in figure 6.

Additionally, the outputs from each consequent were weighted based on their

importance to the classification learned through experience and knowledge of hu-

man experts. The results from each rule were combined using the Max-product

method, and the defuzzification of the data into a crisp output was performed us-

ing the fuzzy-centroid method, which essentially favours the rule with the greatest

area. The system then can take as inputs the values of the spectral features and

have a leveled classification (I, III, or V) with an associated truth-value and the ex-

planation of the system’s reasoning as output. If the truth value for a classification

is significantly small, and alternative classification could be included.

Similar to the other works, this paper also demonstrated the application of

neural networks. In this case, they used backpropagation networks. In addition

to the expert system with fuzzy logic, as described above, they included results

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Figure 8: Taken from [14]. The system performance for spectral types and luminosity.

from an expert system that did not incorporate fuzzy logic. The results can be

seen in figure 7. Additionally the expert system with fuzzy logic performance for

temperature and luminosity is shown in figure 8. The expert systems were able to

classify stars with an error rate below 20%, and errors that were made were usually

explainable. For example, as seen in figure 8, spectral types G, A, and luminosity

type V have high error rates, but the catalog also contained few stars of these types.

More data for these stars would lead to more refined features used and better fuzzy

rules.

Figure 7 shows the overall results. Neural networks are shown to provide better

results for the spectral types and luminosities, whereas expert systems with fuzzy

logic are more suitable for classifying the star’s global temperature. Overall, each

system reached a global success rate of around 90%. It is interesting to note that, in

some cases, the automatic classification techniques performed slightly better than

the two human experts. The results are promising, and future works discussed by

the paper include incorporating the expert system with fuzzy logic and the back-

propagation network to improve results.

6 Improving Astronomical Image Processing

The previously discussed papers all aimed to automate classification of astronomical

images in some way. However, obtaining automatic identification of astronomical

objects is also a challenge that has been exacerbated by the rise of digital imaging

and the vast amount of astronomical data. Questions about the universe, such as

the abundance of Earth-like planets, are being explored with robotic telescopes.

The success largely depends on the accuracy of automated real-time processing of

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images, never seen by humans, to distinguish between known astronomical objects

and new astronomical objects. This is a difficult task itself, as many objects are

extremely faint, objects are moving (such as asteroids or comets), and objects can

change (such as a massive stars turning supernova, and then forming a nebula).

Objects in space change all the time, but this is not the only problem of astro-

nomical imaging. The equipment used can make this problem more challenging as

well. Slight shifts in the orientation of the camera, imperfections in the CCD, and

inaccuracies of the optics are just a few problems that can arise.

Typically, the process for pipeline processing of astronomical images depends

on algorithmic decisions. This requires detecting and isolating single objects in

the image. The first step is usually to locate objects already known in a catalog.

Computing the topocentric coordinates at a specific time of a known star and then

transforming those coordinates to image coordinates would be a simple way to find

the expected locations of any star in the frame. However, as previous mentioned,

other factors make this a difficult challenge.

Shamir and Nemiroff made it their objective to solve this problem in their 2005

paper, [15] (as well as their 2006 paper, [16]). They present an algorithm which uses

fuzzy logic to transform celestial coordinates into (x, y) image coordinates, despite

various noise challenges ( specifically for wide-angle non-linear optical distortions,

slight optical imperfections, and small unrecorded shifts in orientation).

The first step of the algorithm is to manually identify reference stars. It is

important to note that, while this process does involve humans, the task of locating

a few familiar reference stars in a frame is trivial compared to identifying all stars

in the frame. The next step is to build the fuzzy logic models (the rules). The

models are based on the following two transformation functions:

f1 : azimuth 7−→ angle (7)

f2 : altitude, azimuth 7−→ distance (8)

The first rule has one antecedent variable, azimuth, and one consequent variable,

angle. The second rule takes two antecedents, azimuth and altitude, and has

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Figure 9: Taken from [15]. The example membership functions of four fuzzy sets (FS0

to FS1) created for the four example reference stars for equation 7. Assumes that α1 <α2 < α3.

one consequent variable, distance. Azimuth is a measurement in angles from a

reference point (usually North) and the perpendicular projection of the object onto

the horizon. Altitude is the measurement of the height of the object above the

horizon, also measured in degrees. These are easily computed for known stars at

a given time. The angle and distance are both measurements of the camera, in

degrees and pixels (respectively).

For each reference star, a fuzzy set and a fuzzy rule is added to the model.

Figure 9 is an example of the fuzzy logic model for equation 7 using four reference

stars with (azimuth, altitude, angle, and distance) of (0, ε0, θ0, R0), (α1, ε1, θ1, R1),

(α2, ε2, θ2, R2), and (α3, ε3, θ3, R3). In general, membership functions are built as

triangles that peak at the reference value, and intersects the x-axis at the reference

value of its neighboring points. Because of this, almost all azimuth values belong

to two fuzzy sets (the exception are those at the maximum of a particular fuzzy

set). Additionally, each reference star i adds the rule

FSi 7−→ θi (9)

To compute the angle for any other stellar object, it is simply a matter of computing

the membership value for each fuzzy set. The results of the rules are then combined

using a weighted average defuzzification method. The results are similar to those

of a linear interpolation [15]. The technique for building the model of equation 8 is

similar.

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This fuzzy logic based transformation algorithm has been tested and is in prac-

tical use with the Night Sky Live project [13]. The Night Sky Live project uses

fish-eye lens cameras to take images of the entire night sky, which are then analyzed.

The data is made freely available for scientific or public use. It continuously tracks

the objects in the night sky, and any non-cataloged bright objects are immediately

detected.

Before this algorithm was used, a previous Night Sky Live identification algo-

rithm employed a straight forward analytic transformation and was only accurate

for stars with a magnitude of about 3.5 in best cases. The current algorithm has

dramatically improved the identification and has practically 100 percent accuracy

for identification of stars down to a magnitude of 5.6. To put this in perspective,

magnitudes measure the brightness of stellar objects. The more positive the mag-

nitude, the more faint the object is. The sun is currently the brightest object in

the sky, with a magnitude of about -27. The full moon has a magnitude around

-13. The brightest star in the night sky, Sirius, has a magnitude of -1.4. Objects

with magnitudes higher than 6 are no longer visible to the naked eye.

These are obviously exciting results. For future works, the authors aim to

improve the accuracy for stellar objects with even fainter magnitudes. The Night

Sky Live project’s equipment can currently only produce images of objects brighter

than a magnitude of 6.8, so to improve the algorithm to include these faint objects

would be a first step. Also, it would be fascinating to see how this technique works

with deep and ultra-deep field telescopes, such as the Hubble telescope, which have

been able to take images of objects created soon after the big bang [6].

7 Discussion and Conclusion

The need for an automated data mining system for astronomical data is clearly ap-

parent. However, the task does not seem to be as straight forward as simply creating

an algorithm that will classify any celestial object. Many papers have attempted

to tackle smaller portions of the problem. As discussed in this paper, various works

have attempted to use fuzzy logics, including fuzzy similarity, fuzzy inference, and

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fuzzy clustering algorithms, to tackle the problem of star and galaxy separation

([5], [12], [11]). Other works attempt to choose even further sub-problems, such

as classifying specific types of object. [14] attempted to do this with stars. The

results demonstrate that it is possible to create automatic classification techniques

for stars that perform as well as human experts

Additionally, improving classification algorithms is not the only way people have

tried to improve the data mining of the astronomical data constantly pouring in.

Works, such as [15] and [16], aimed to improve the images obtained by telescopes.

In this way, classification techniques would have less interference from noise due

to shifts in orientation, wide-angle non-linear optical distortions, and alight optical

perfections.

There are no known techniques that have the ability to be as good as human

observers when it comes to general classification, but there are many techniques

that do very well at solving the sub-problems of classification. Additionally, fuzzy

logic has been shown to produce reasonable results when compared to non-fuzzy

logic algorithms counterparts and have also been put to practical use. While using

fuzzy logic alone might not be the optimal solution, integrating fuzzy logic as a

preclassifier or using it for improved astronomical image processing could lead to

an overall optimal result. Current applications of fuzzy logic are promising, and

also open the door to many new future works and applications. There remains

much to be studied about the applications and practicality of fuzzy logic and its

employment for astronomical data mining.

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References

[1] Borucki, W.J. et al. ”Kepler Mission: a mission to find Earth-size planets inthe habitable zone.” Proceedings of the Conference on Towards Other Earths.(2003): 69–81.

[2] Clampin, M. et al. ”Status of the JWST observatory design.” American As-tronomical Society Meeting 203, 121.01. 35 (2003): 1408.

[3] Donald G. York et al. ”The sloan digital sky survey: technical summary.”Astronomical Journal. 120.3 (2000): 1579-87.

[4] Fukugita, M. et al. ”A catalog of morphologically classified galaxies from thesloan digital sky survey: North equato-rial region.” The Astronomical Journal,134 (2007): 579-593.

[5] Gauci, Adam, Kristian Zarb Adami, John Abela. ”Machine Learning forGalaxy Morphology Classification” Computer. 8 (2010): 1-8.

[6] ”Hubble’s Deepest View Ever of the Universe Unveils Earli-est Galaxies”. March 9, 2004. NASA. Retrieved 2008-12-27.<http://hubblesite.org/newscenter/archive/releases/2004/07/>

[7] Kaiser, Nicholas. ”Pan-STARRS: a wide-field optical survey telescope array.”Proceedings of the SPIE. 5489 (2004): 11-22.

[8] Kohonen, T. Self-Organizing Maps. Berlin: Springer, 1995.

[9] Kurtz, M. J. The MK process and stellar classification of stars. Toronto: DavidDunlap Observatory, 1984.

[10] Lintott, Chris J. et al. ”Galaxy Zoo: morphologies derived from visual inspec-tion of galaxies from the Sloan Digital Sky Survey”. Monthly Notices of theRoyal Astronomical Society. 389.3 (2008): 1179-1189.

[11] Longo, G et al. “Advanced data mining tools for exploring large astronomicaldatabases.” Proceedings of SPIE. 4477 (2001): 61-75.

[12] Mahonen, Petri, Tapio Frantti. ”Fuzzy classifier for star-galaxy separation.”The Astrophysical Journal. 541.1 (2000): 261-263.

[13] Nemiroff, R.J and L. Shamir. ”Toward a Continuous Record of the Sky.” As-tronomical Society of the Pacific. 3.761 (1999): 886-897.

[14] Rodrıguez, Alejandra, Bernardino Arcaya, Carlos Dafontea, Minia Manteigab,and Iciar Carricajob. ”Automated knowledge-based analysis and classificationof stellar spectra using fuzzy reasoning.” Expert Systems with Applications.27.2 (2004): 237–244.

[15] Shamir, Lior, and Robert J Nemiroff. “A fuzzy logic based algorithm for findingastronomical objects in wide-angle frames.” Publications of the AstronomicalSociety of Australia. 22 (2005): 111-117.

[16] Shamir, Lior, and Robert J Nemiroff. ”Astronomical pipeline processing usingfuzzy logic.” Applied Soft Computing. 8.1 (2006): 79-87.

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